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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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by Davar Khoshnevisan and Yimin Xiao PDF
Trans. Amer. Math. Soc. 359 (2007), 3125-3151 Request permission

Abstract:

An $N$-parameter Brownian sheet in $\mathbf {R}^d$ maps a non-random compact set $F$ in $\mathbf {R}^N_+$ to the random compact set $B(F)$ in $\mathbf {R}^d$. We prove two results on the image-set $B(F)$: (1) It has positive $d$-dimensional Lebesgue measure if and only if $F$ has positive $\frac d 2$-dimensional capacity. This generalizes greatly the earlier works of J. Hawkes (1977), J.-P. Kahane (1985), and Khoshnevisan (1999). (2) If $\dim _{_\mathcal {H}}F > \frac d 2$, then with probability one, we can find a finite number of points $\zeta _1,\ldots ,\zeta _m\in \mathbf {R}^d$ such that for any rotation matrix $\theta$ that leaves $F$ in $\mathbf {R}^N_+$, one of the $\zeta _i$’s is interior to $B(\theta F)$. In particular, $B(F)$ has interior-points a.s. This verifies a conjecture of T. S. Mountford (1989). This paper contains two novel ideas: To prove (1), we introduce and analyze a family of bridged sheets. Item (2) is proved by developing a notion of “sectorial local-non-determinism (LND).” Both ideas may be of independent interest. We showcase sectorial LND further by exhibiting some arithmetic properties of standard Brownian motion; this completes the work initiated by Mountford (1988).
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Additional Information
  • Davar Khoshnevisan
  • Affiliation: Department of Mathematics, The University of Utah, 155 S. 1400 E., Salt Lake City, Utah 84112–0090
  • MR Author ID: 302544
  • Email: davar@math.utah.edu
  • Yimin Xiao
  • Affiliation: Department of Statistics and Probability, A-413 Wells Hall, Michigan State University, East Lansing, Michigan 48824
  • Email: xiao@stt.msu.edu
  • Received by editor(s): September 12, 2004
  • Received by editor(s) in revised form: April 21, 2005
  • Published electronically: February 14, 2007
  • Additional Notes: This research was supported by a generous grant from the National Science Foundation
  • © Copyright 2007 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Trans. Amer. Math. Soc. 359 (2007), 3125-3151
  • MSC (2000): Primary 60G15, 60G17, 28A80
  • DOI: https://doi.org/10.1090/S0002-9947-07-04073-1
  • MathSciNet review: 2299449